Inference for Stereological Extremes

In the production of clean steels the occurrence of imperfections — so-called inclusions — is unavoidable. Furthermore, the strength of a clean steel block is largely dependent on the size of the largest imperfection it contains, so inference on extreme inclusion size forms an important part of quality control. Sampling is generally done by measuring imperfections on planar slices, leading to an extreme value version of a standard stereological problem: how to make inference on large inclusions using only the sliced observations. Under the assumption that inclusions are spherical, this problem has previously been tackled using a combination of extreme value models, stereological calculations, a Bayesian hierarchical model and standard Markov chain Monte Carlo (MCMC) techniques. Our objectives in this article are two-fold: to assess the robustness of such inferences with respect to the assumption of spherical inclusions, and to develop an inference procedure that is valid for non-spherical inclusions. We investigate both of these aspects by extending the spherical family for inclusion shapes to a family of ellipsoids. The issue of robustness is then addressed by assessing the performance of the spherical model when fitted to measurements obtained from a simulation of ellipsoidal inclusions. The issue of inference is more difficult, since likelihood calculation is not feasible for the ellipsoidal model. To handle this aspect we propose a modification to a recently developed likelihood-free MCMC algorithm. After verifying the viability and accuracy of the proposed algorithm through a simulation study, we analyze a real inclusion dataset, comparing the inference obtained under the ellipsoidal inclusion model with that previously obtained assuming spherical inclusions.